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In the name of GodIn the name of God

Network Flows

1. Introduction

1.2 Notation and Definitions1.2 Notation and Definitions

Fall 2010Instructor: Dr. Masoud Yaghini

Notation and Definitions

� In this section we give several basic definitions from

graph theory and present some basic notation.

� We also state some elementary properties of graphs.


� Directed graph

– A directed graph G = (N, A) consists of a set N of nodes and

a set A of arcs whose elements are ordered pairs of distinct

nodes.

– N = {l, 2, 3, 4, 5, 6, 7}

– A = {(l, 2), (1, 3), (2, 3), (2,4), (3, 6), (4, 5), (4, 7), (5, 2),

(5, 3), (5, 7), (6, 7)}.


� Directed network

– A directed network is a directed graph whose nodes

and/or arcs have associated:

� costs

� capacities

� and/or supplies and demands� and/or supplies and demands

� We use the terms “graph” and “network”

synonymously.

� We let n denote the number of nodes and m denote the

number of arcs in G.


� Undirected Graphs

– In an undirected graph arcs are unordered pairs of distinct

nodes.

– In an undirected graph, we can refer to an arc joining the

node pair i and j as either (i, j) or (j, i).


� Tails and Heads

– A directed arc (i, j) has two endpoints i and j.

– Node i is the tail of arc (i, j) and node j is its head.

– The arc (i, j) emanates from node i and terminates at node

j.

– An arc (i, j) is incident to nodes i and j. – An arc (i, j) is incident to nodes i and j.

– The arc (i, j) is an outgoing arc of node i and an incoming

arc of node j.

– Whenever an arc (i, j) œ A, we say that node j is adjacent to

node i.


� Degrees

– The indegree of a node is the number of incoming arcs of

that node and its outdegree is the number of its outgoing

arcs.

– The degree of a node is the sum of its indegree and

outdegree. outdegree.

– The sum of indegrees of all nodes equals the sum of

outdegrees of all nodes and both are equal to the number of

arcs m in the network.


� Example of degree:

– Node 3 has an indegree of 3, an outdegree of 1, and a

degree of 4.


� Adjacency List

– The arc adjacency list A(i) of a node i is the set of arcs

emanating from that node, A(i) = {(i, j) œ A: j œ N}.

– The node adjacency list A(i) is the set of nodes adjacent to

that node; in this case, A(i) = {j œ N: (i, j) œ A}.

– We assume that arcs in the adjacency list A(i) are arranged – We assume that arcs in the adjacency list A(i) are arranged

so that the head nodes of arcs are in increasing order.


� Multiarcs and Loops

– Multiarcs are two or more arcs with the same tail and head

nodes.

– A loop is an arc whose tail node is the same as its head

node.


� Subgraph

– A graph G' = (N', A') is a subgraph of G = (N, A) if N' Œ N

and A' Œ A.

– We say that G' = (N', A') is the subgraph of G induced by

N' if A' contains each arc of A with both endpoints in N'.

– A graph G' = (N', A') is a spanning subgraph of G = (N, A) – A graph G' = (N', A') is a spanning subgraph of G = (N, A)

if N' = N and A' Œ A.


� Walk

– A walk in a directed graph G = (N, A) is a subgraph of G

consisting of a sequence of nodes and arcs

i1 – a1 - i2 - a2 - ... – ir-1 – ar-1 - ir

– satisfying the property that for all 1 § k § r - 1,

– either ak = (ik , ik+1) œ A or ak = (ik+1 , ik) œ A. – either ak = (ik , ik+1) œ A or ak = (ik+1 , ik) œ A.

– Alternatively, we shall sometimes refer to a walk as a set of

(sequence of) arcs (or of nodes) without any explicit

mention of the nodes (without explicit mention of arcs).


� (a) 1-2-5-7

� (b) 1-2-4-5-2-3


� Directed Walk

– A directed walk is an oriented version of a walk in the

sense that for any two consecutive nodes ik and ik+1 on the

walk, (ik , ik+1) œ A.

– (a) is not a directed walk

– (b) is a directed walk


� Path

– A path is a walk without any repetition of nodes.

– Directions are ignored.

– (a) is a path, 1-2-5-7

– (b) is not a path because it repeats node 2 twice., 1-2-4-5-2-

3


� We can partition the arcs of a path into two groups:

forward arcs and backward arcs.

� The arcs (1, 2) and (5, 7) are forward arcs and the arc

(5, 2) is a backward arc.


� Directed Path

– A directed path is a directed walk without any repetition of

nodes.

– In other words, a directed path has no backward arcs.

– Directions are important.


� Cycle

– A cycle is a path i1 - i2 - ... - ir together with the arc (ir, i1) or

(i1, ir).

– We refer to a cycle using the notation i1- i2 - ... - ir – i1.

– Just as we did for paths, we can define forward and

backward arcs in a cycle. backward arcs in a cycle.

– Directions are ignored.

– The arcs (5, 3) and (3, 2) are forward arcs and the arc (5, 2)

is a backward arc of the cycle 2-5-3.


� Directed Cycle

– A directed cycle is a directed path i1 - i2 - … - ir together

with the arc (ir, i1).

– Directions are important.

– (a) is a cycle, but not a directed cycle;

– (b) is a directed cycle.


� Acyclic Graph

– A graph is a acyclic if it contains no directed cycle.


� Connectivity

– We will say that two nodes i and j are connected if the

graph contains at least one path from node i to node j.

– A graph is connected if every pair of its nodes is connected;

otherwise, the graph is disconnected.

– (a) is connected graph, and (b) is disconnected graph


� Components

– We refer to the maximal connected subgraphs of a

disconnected network as its components.

– This graph has two components consisting of the node sets

{l, 2, 3, 4} and {5, 6}.


� Strong Connectivity

– A connected graph is strongly connected if it contains at

least one directed path from every node to every other

node.

– A strongly connected graph


� Cut

– A cut is a partition of the node set N into two parts, S and

S̅ = N - S.

– Each cut defines a set of arcs consisting of those arcs that

have one endpoint in S and another endpoint in S̅.

– We refer to this set of arcs as a cut and represent it by the – We refer to this set of arcs as a cut and represent it by the

notation [S, S̅].


– A cut with S = {l, 2, 3} and S̅ = {4, 5, 6, 7}.

– The set of arcs in this cut are {(2, 4), (5, 2), (5, 3), (3, 6)}.


� s-t Cut

– An s-t cut is defined with respect to two distinguished

nodes s and t, and is a cut [S, S̅] satisfying the property that

s œ S and t œ S̅.

– If s = 1 and t = 6, this cut is an s-t cut;

– but if s = 1 and t = 3, this cut is not an s-t cut.


� Tree

– A tree is a connected graph that contains no cycle.

� Property

– (a) A tree on n nodes contains exactly n - 1 arcs.

– (b) A tree has at least two leaf nodes (i.e., nodes with

degree 1). degree 1).

– (c) Every two nodes of a tree are connected by a unique

path.


� Two examples of trees


� Forest

– A graph that contains no cycle is a forest.

– Alternatively, a forest is a collection of trees.

– an example of a forest.


� Subtree

– A connected subgraph of a tree is a subtree.


� Rooted Tree

– A rooted tree is a tree with a specially designated node,

called its root; we regard a rooted tree as though it were

hanging from its root.

– an instance of a rooted tree; node 1 is the root node.


� We often view the arcs in a rooted tree as defining

predecessor-successor (or parent-child) relationships.

– Node 5 is the predecessor of nodes 6 and 7,

– Node 1 is the predecessor of nodes 2, 4, and 5.


� Each node i (except the root node) has a unique

predecessor, which is the next node on the unique path

in the tree from that node to the root;

� We store the predecessor of node i using a predecessor

index pred(i).

If j = pred(i), we say that node j is the predecessor of � If j = pred(i), we say that node j is the predecessor of

node i and node i is a successor of node j.

� These predecessor indices uniquely define a rooted

tree and also allow us to trace out the unique path from

any node back to the root.


� There are two special type of rooted trees, called a

directed in-tree and a directed out-tree.


� Directed-Out- Tree

– A tree is a directed out-tree routed at node s if the unique

path in the tree from node s to every other node is a directed

path.

– an instance of a directed out-tree rooted at node 1.

– Observe that every node in the directed outtree (except

node 1) has indegree 1.


� Directed-In-Tree

– A tree is a directed in-tree routed at node s if the unique

path in the tree from any node to node s is a directed path.

– an instance of a directed in-tree rooted at node 1.

– Observe that every node in the directed in-tree (except node

1) has outdegree 1.


� Spanning Tree

– A tree T is a spanning tree of G if T is a spanning subgraph

of G.

– Every spanning tree of a connected n-node graph G has (n -

1) arcs.

– We refer to the arcs belonging to a spanning tree T as tree – We refer to the arcs belonging to a spanning tree T as tree

arcs and arcs not belonging to T as nontree arcs.


– two spanning trees of the graph


� Undirected networks

– An undirected arc (i, j) has two endpoints, i and j, but its

tail and head nodes are undefined.

– If the network contains the arc (i, j), node i is adjacent to

node j, and node j is adjacent to node i.

The End